Given a covector ω\omega at aa and a tangent vectorvv at aa, the pairing ⟨ω,v⟩\langle{\omega,v}\rangle is a scalar (a real number, usually). This (with some details about linearity and universality) is basically what it means for T*(X)T^*(X) to be dual to T*(X)T_*(X). More globally, given a covector field ω\omega and a tangent vector fieldvv, the paring ⟨ω,v⟩\langle{\omega,v}\rangle is a scalar function on XX.

Given a point aa in XX and a differentiable (real-valued) partial functionff defined near aa, the differentialdaf\mathrm{d}_a f of ff at aa is a covector on XX at aa; given a tangent vector vv at aa, the pairing is given by

⟨daf,v⟩=v[f], \langle{\mathrm{d}_a f, v}\rangle = v[f] ,

thinking of vv as a derivation on differentiable functions defined near aa. (It is really the germ at aa of ff that matters here.) More globally, given a differentiable function ff, the differentialdf\mathrm{d}f of ff is a covector field on XX; given a vector field vv, the pairing is given by

⟨df,v⟩=v[f], \langle{\mathrm{d}f, v}\rangle = v[f] ,

thinking of vv as a derivation on differentiable functions.

One can also define covectors at aa to be germs of differentiable functions at aa, modulo the equivalence relation that daf=dag\mathrm{d}_a f = \mathrm{d}_a g if f−gf - g is constant on some neighbourhood of aa. In general, a covector field won't be of the form df\mathrm{d}f, but it will be a sum of terms of the form hdfh \mathrm{d}f. More specifically, a covector field ω\omega on a coordinate patch can be written

ω=∑iωidxi \omega = \sum_i \omega_i\, \mathrm{d}x^i

in local coordinates (x1,…,xn)(x^1,\ldots,x^n). This fact can also be used as the basis of a definition of the cotangent bundle.